1549380323-Statistical Mechanics Theory and Molecular Simulation

Simple examples 383

wave in the box. This is possible because the Schr ̈odinger equation islinear, hence any
linear combination of eigenfunctions with the same eigenvalue is also aneigenfunction.
In this case, we need to take

ψp(x) =Csin

(px

̄h

)

=

C

2 i

[

eipx/ ̄h−e−ipx/h ̄

]

, (9.3.15)

which manifestly satisfies the boundary condition atx= 0. This function satisfies
the boundary condition atx=Lonly ifpL/ ̄h=nπ, wherenis a positive integer.
This leads to the momentum quantization conditionp=nπL/ ̄h≡pnand the energy
eigenvalues

En=

p^2 n

2 m

=

̄h^2 π^2

2 mL^2

n^2. (9.3.16)

The eigenfunctions become

ψn(x) =Csin

(nπx

L

)

. (9.3.17)

Normalizing yieldsC=

√

2 /Lfor the constant. From eqn. (9.3.17), it is clear whyn
must be strictly positive. Ifn= 0, thenψn(x) = 0 everywhere, which would imply that
the particle exists nowhere. Finally, since the eigenfunctions are already constructed
from combinations of right- and left-propagating waves, to form standing waves in the
box, allowingn <0 only changes the sign of the eigenfunctions (which is a trivial
phase factor) but not the physical content of the eigenfunctions (probabilities and
expectation values are not affected by an overall sign). Note thatthe probability
distributionPn(x) = (2/L) sin^2 (nπx/L) is no longer uniform.

9.3.2 The harmonic oscillator

The second example we will consider is a single one-dimensional particlemoving in a
harmonic potentialU(x) =mω^2 x^2 /2, so that the Hamiltonian becomes

Hˆ= pˆ

2

2 m

+

1

2

mω^2 xˆ^2. (9.3.18)

The eigenvalue equation forHˆbecomes, according to eqn. (9.2.52),

[

−

̄h^2

2 m

d^2

dx^2

+

1

2

mω^2 x^2

]

ψn(x) =Enψn(x). (9.3.19)

Here, we have anticipated that because the particle is asymptotically bound (U(x)→
∞asx→±∞), the energy eigenvalues will be discrete and characterized by an integer
n. Since the potential becomes infinitely large asx→ ±∞, we have the boundary
conditionsψn(∞) =ψn(−∞) = 0.
The solution of this second-order differential equation is not trivialand, therefore,
we will not carry out its solution in detail. However, the interested reader is referred
to the excellent treatment inPrinciples of Quantum Mechanicsby R. Shankar (1994).